Abstract
A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r − 1)2 ≤ δ + \( \sqrt \delta \) − 2 < r 2 and all (δ, g)-cages with even girth g ≥ 10 are r-connected, where r is the largest integer satisfying \( \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta \). These results support a conjecture of Fu, Huang and Rodger that all (δ, g)-cages are δ-connected.
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Supported by 973 Project (2006CB805904) of Ministry of Science and Technology of China, and Discovery Grant (144073) of Natural Sciences and Engineering Research Council of Canada
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Lu, H.L., Wu, Y.J., Yu, Q.L. et al. New improvements on connectivity of cages. Acta. Math. Sin.-English Ser. 27, 1163–1172 (2011). https://doi.org/10.1007/s10114-011-8279-8
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DOI: https://doi.org/10.1007/s10114-011-8279-8